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Chapter 8

- Similarity

8.1

- Ratio and Proportion

Ratios

- Ratio- Comparison of 2 quantities in the same

units - The ratio of a to b can be written as
- a/b
- a b
- The denominator cannot be zero

Simplifying Ratios

- Ratios should be expressed in simplified form
- 68 34
- Before reducing, make sure that the units are the

same. - 12in 3 ft
- 12in 36 in
- 1 3

Examples (page 461)

- Simplify each ratio
- 10. 16 students
- 24 students
- 12. 22 feet
- 52 feet
- 18. 60 cm
- 1 m

Examples (page 461)

- Simplify each ratio
- 20. 2 mi
- 3000 ft
- 24. 20 oz.
- 4 lb
- There are 5280 ft in 1 mi.
- There are 16 oz in 1 lb.

Examples (page 461)

- Find the width to length ratio
- 14.
- 16.

Using Ratios Example 1

- The perimeter of the isosceles triangle shown is

56 in. The ratio of LM MN is 54. Find the

length of the sides and the base of the triangle.

Using Ratios Example 2

- The measures of the angles in a triangle are in

the extended ratio 348. Find the measures of

the angles

4x

8x

3x

Using Ratios Example 3

- The ratios of the side lengths of ?QRS to the

corresponding side lengths of ?VTU are 32. Find

the unknown lengths.

2 cm

18 cm

Proportions

- Proportion
- Ratio Ratio
- Fraction Fraction
- Means and Extremes
- Extreme Mean Mean Extreme

Solving Proportions

- Solving Proportions
- Cross multiply
- Let the means equal the extremes
- Example

Properties of Proportions

- Cross Product Property
- Reciprocal Property

Solving Proportions Example 1

Solving Proportions Example 2

Solving Proportions Example 3

- A photo of a building has the measurements shown.

The actual building is 26 ¼ ft wide. How tall

is it?

2.75 in

1 7/8 in

8.2

- Problem solving in Geometry with Proportions

Properties of Proportions

Example 1

- Tell whether the statement is true or false
- A.
- B.

Example 2

- In the diagram
- Find the length of LQ.

Geometric Mean

- Geometric Mean
- The geometric mean between two numbers a and b is

the positive number x such that - ex 8/4 4/2

Example 3

- Find the geometric mean between 4 and 9.

Similar Polygons

- Polygons are similar if and only if
- the corresponding angles are congruent
- and
- the corresponding sides are proportionate.

- Similar figures are dilations of each other.

(They are reduced or enlarged by a scale factor.) - The symbol for similar is ?

Example 1

Determine if the sides of the polygon are

proportionate.

Example 2

Determine if the sides of the polygon are

proportionate.

Example 3

Find the missing measurements. HAPIE ? NWYRS

AP EI SN YR

Example 4

Find the missing measurements. QUAD ? SIML

QD MI m?D

m?U m?A

8.4/8.5

- Similar Triangles

Similar Triangles

- To be similar, corresponding sides must be

proportional and corresponding angles are

congruent.

Similarity Shortcuts

- AA Similarity Shortcut
- If two angles in one triangle are congruent to

two angles in another triangle, then the

triangles are similar.

Similarity Shortcuts

- SSS Similarity Shortcut
- If three sides in one triangle are proportional

to the three sides in another triangle, then the

triangles are similar.

Similarity Shortcuts

- SAS Similarity Shortcut
- If two sides of one triangle are proportional to

two sides of another triangle and - their included angles are congruent, then the

triangles are similar.

Similarity Shortcuts

- We have three shortcuts
- AA
- SAS
- SSS

Example 1

Example 2

Example 3

- 4. A flagpole 4 meters tall casts a 6 meter

shadow. At the same time of day, a nearby

building casts a 24 meter shadow. How tall is

the building?

- 5. Five foot tall Melody casts an 84 inch

shadow. How tall is her friend if, at the same

time of day, his shadow is 1 foot shorter than

hers?

- 6. A 10 meter rope from the top of a flagpole

reaches to the end of the flagpoles 6 meter

shadow. How tall is the nearby football goalpost

if, at the same moment, it has a shadow of 4

meters?

- 7. Private eye Samantha Diamond places a mirror

on the ground between herself and an apartment

building and stands so that when she looks into

the mirror, she sees into a window. The mirror

is 1.22 meters from her feet and 7.32 meters from

the base of the building. Sams eye is 1.82

meters above the ground. How high is the window?

1.22

7.32

8.6

- Proportions and Similar Triangles

Proportions

- Using similar triangles missing sides can be

found by setting up proportions.

Theorem

- Triangle Proportionality Theorem
- If a line parallel to one side of a triangle

intersects the other two sides, then it divides

the two sides proportionally.

Theorem

- Converse of the Triangle Proportionality Theorem
- If a line divides two sides of a triangle

proportionally, then it is parallel to the third

side.

Example 1

- In the diagram, segment UY is parallel to segment

VX, UV 3, UW 18 and XW 16. What is the

length of segment YX?

Example 2

- Given the diagram, determine whether segment PQ

is parallel to segment TR.

Theorem

- If three parallel lines intersect two

transversals, then they divide the transversals

proportionally.

Theorem

- If a ray bisects an angle of a triangle, then it

divides the opposite side into segments whose

lengths are proportional to the lengths of the

other two sides.

Example 3

- In the diagram, ?1 ? ?2 ? ?3, AB 6, BC9, EF8.

What is x?

Example 4

- In the diagram, ?LKM ? ?MKN. Use the given side

lengths to find the length of segment MN.

- 5. Juanita, who is 1.82 meters tall, wants to

find the height of a tree in her backyard. From

the trees base, she walks 12.20 meters along the

trees shadow to a position where the end of her

shadow exactly overlaps the end of the trees

shadow. She is now 6.10 meters from the end of

the shadows. How tall is the tree?

8.7

- Dilations

Dilations

- Dilation Transformation that maps all points so

that the proportion stands true. - Enlargement A dilation which makes the

transformed image larger than the original image - Reduction A dilation which makes the transformed

image smaller than the original image.

Enlargement

An enlargement has a scale factor of k which if

found by the proportion . In an

enlargement k is always greater than 1.

Find k

Reduction

- A reduction has a scale factor of k which is

found by the proportion . In a reduction, 0 lt

k lt 1.

C

6

P

P

14

Find k

Dilations in a coordinate plane

- If the center of the dilation is the origin, the

image can be found by multiplying each coordinate

by the scale factor - Example
- Original coordinates
- (3, 6), (6, 12) and (9, 3)
- Scale factor 1/3
- Find the image coordinates.